An ant colony hyperheuristic approach for matrix bandwidth reduction

In this chapter we come to grips with the question of how to solve, in practice, systems of linear equations. Our strategy will be to gather a certain number of observations which will allow us to form a strategy. Given that many mathematicians often use the adjective clear for something which usually is far from clear but is tiresome to prove we will prove our good intention to avoid this bad habit by beginning with an observation that is totally clear.

Solutions of Systems of Linear Equations

This paper proposes a new framework, named Deja-Vu+, which is an extension of Deja Vu framework, a classic study on hyper-heuristic framework with 2R (Record and Recall) modules. Deja-Vu+ has the ability to handle two other domains, namely regression and unsupervised learning. The extension examines the strength of Deja-Vu+ for solving regression and unsupervised learning tasks. The regression problems are treated here as multiclass classification tasks, and unsupervised learning tasks are considered as clustering problems. The proposed framework is tested on a number of regression and unsupervised learning benchmark problems and has shown promising results to handle regression as classification. The framework attains an overall average accuracy of 70% for regression and clustering data sets. Deja-Vu+ is knowledge-rich hyper-heuristic framework, which is capable enough to transfer knowledge successfully. This knowledge transfer improves the performance of learning by avoiding the extensive heuristic search process. Our experimental results show that using previously attained knowledge to reduce the computational effort is beneficial in solving multi-disciplinary machine learning problems.

Memory augmented hyper-heuristic framework to solve multi-disciplinary problems inspired by cognitive problem solving skills

Researchers used graph-theory approaches to design the state-of-theart low-cost heuristics for profile reduction. This paper evolves and selects four low-cost heuristics for profile reduction using a genetic programming hyperheuristic approach. This paper evaluates the resulting heuristics for profile reduction from the genetic programming hyperheuristic approach in two application areas against the low-cost heuristics for solving the problem. The results obtained on a set of standard benchmark matrices taken from the SuiteSparse sparse matrix collection indicate that the resulting heuristics from the genetic programming hyperheuristic approach does not compare favorably with a high-quality heuristics for profile reduction.

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An experimental analysis of a GP hyperheuristic approach for evolving low-cost heuristics for profile reductions

The bandwidth minimization problem is a well-known \(\mathcal {NP}\)-hard problem. This paper describes our experience in implementing a biased random-key genetic algorithm for the bandwidth reduction problem. Specifically, this paper compares the results of the new algorithm with the results yielded by four approaches. The results obtained on a set of standard benchmark matrices taken from the SuiteSparse sparse matrix collection indicated that the novel approach did not compare favorably with the state-of-the-art metaheuristic algorithm for bandwidth reduction. The former seems to be faster than the latter. On the other hand, the design of heuristics for bandwidth reduction is a very consolidated research area. Thus, a paradigm shift seems necessary to design a heuristic with better results than the state-of-the-art meta-heuristic algorithm at shorter execution times.

A Biased Random-Key Genetic Algorithm for Bandwidth Reduction

This paper proposes a novel ant colony hyperheuristic approach for reordering the rows and columns of symmetric positive definite matrices. This ant colony hyperheuristic approach evolves heuristics for bandwidth reduction applied to instances arising from specific application areas with the objective of generating low-cost reordering algorithms. This paper evaluates the resulting reordering algorithm in each application area against state-of-the-art reordering algorithms with the purpose of reducing the running times of the zero-fill incomplete Cholesky-preconditioned conjugate gradient method. The results obtained on a wide-ranging set of standard benchmark matrices show that the proposed approach compares favorably with state-of-the-art reordering algorithms when applied to instances arising from computational fluid dynamics, structural, and thermal problems.

Evolving reordering algorithms using an ant colony hyperheuristic approach for accelerating the convergence of the ICCG method

The solution of linear systems represented by Ax = b is fundamental in many numerical simulations in science and engineering. Reducing the profile of A can reduce the storage requirements and time processing costs of solving such linear systems. In this work, we propose a generalized algorithm for finding pseudo–peripheral vertices for Snay’s heuristic. In experiment performed on 36 instances contained in the Harwell-Boeing and SuiteSparse matrix collections, it has been found that the number of pseudo– peripheral vertices selected in Snay’s heuristic may be suitable for small instances, but it is insufficient to obtain reasonable results in instances that are not small. This paper recommends to select up to 26% (0.3%) of pseudo–peripheral vertices in relation to the instance size when applied to instances smaller than 3,000 (larger than 20,000) vertices.

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A Novel Approach to Find Pseudo–peripheral Vertices for Snay’s Heuristic

This paper considers the bandwidth reduction problem for large-scale sparse matrices in serial computations. A heuristic for bandwidth reduction reorders the rows and columns of a given sparse matrix. Thus, the method places entries with a nonzero value as close to the main diagonal as possible. Bandwidth optimization is a critical issue for many scientific and engineering applications. This manuscript proposes two heuristics for the bandwidth reduction of large-scale matrices. The first is a variant of the Fast Node Centroid Hill-Climbing algorithm, and the second is an algorithm based on the iterated local search metaheuristic. This paper then experimentally compares the solutions yielded by the new reordering algorithms with the bandwidth solutions delivered by state-of-the-art heuristics for the problem, including tests on large-scale problem matrices. A considerable number of results for a range of realistic test problems showed that the performance of the two new algorithms compared favorably with state-of-the-art heuristics for bandwidth reduction. Specifically, the variant of the Fast Node Centroid Hill-Climbing algorithm yielded the overall best bandwidth results.

S. L. Gonzaga de Oliveira

Papers

Evolving reordering algorithms using an ant colony hyperheuristic approach for accelerating the convergence of the ICCG method

An ant colony hyperheuristic approach for matrix bandwidth reduction

A Novel Approach to Find Pseudo–peripheral Vertices for Snay’s Heuristic

A Biased Random-Key Genetic Algorithm for Bandwidth Reduction

Metaheuristic algorithms for the bandwidth reduction of large-scale matrices